# Is there an accepted axiomatic approach to general relativity?

I am reading Steven Weinberg's book Gravitation and Cosmology. He makes a big deal out of the equivalence principle and showed a bunch of deductions you can make based on it. This surprised me since other books I have read haven't emphasized it as much.

My Question:

Is there an accepted set of axioms or principles that constitute the core premises of GR from which many, most or all relevant properties can be deduced?

• You can write down the axioms of Pseudo-Riemannian geometry and identify some of the mathematical quantities with up-in-the-air words like "space", "time", "observer trajectory", etc. I don't think you need the equivalence principle for this shut up an calculate approach, but you can see it realized in Einsteins theory, if you bend your words correctly. Historically, the principle is more of a guiding principle. I'm not sure to what extent it can hold exactly in GR if you furnish your space with bodies which are all of finite width, so that 1d trajectories don't actually describe body paths. Commented Feb 11, 2015 at 19:56
• Also, I've seen this paper being linked here, which might contain more info: Especially on how the ideas and notions evolved: General covariance and the foundations of general relativity - eight decades of dispute. These two now got long for a comment, please somebody tell me when I should remove it, after OP got the paper link. Commented Feb 11, 2015 at 20:00
• Axioms!?! This isn't the Math StackExchange! :)
– Paul
Commented Feb 11, 2015 at 20:03
• Note that axioms in the sense of Weinberg is not a rigorous interpretation of the word "axiom". Weinberg also "axiomatically" constructs QFT from Lorentz invariance and the cluster decomposition principle, but his construction is not exactly what one would conventionally call axiomatic QFT. Commented Feb 11, 2015 at 20:26
• @ACuriousMind That seems like a casual use of the word "axiom" then, at least compared to a mathematician. Commented Feb 11, 2015 at 20:29

General relativity can be constructed from the following principles:

1. The Principle of Equivalence

2. Vanishing torsion assumption ($$\nabla_XY-\nabla_YX=[X,Y]$$)

3. The Poisson equation (or any other equivalent Newtonian mechanics equation)

Explanations:

1. The Equivalence Principle can be used to show that spacetime is locally Minkowskian, i.e. the laws of special relativity hold in an infinitesimal region around a freely-falling observer. This is equivalent to the mathematical idea that a manifold of dimension $$n$$ is locally homeomorphic to $$\mathbb{R}^n$$. This allows to do two things (that I can think of at the moment). We conclude that spacetime is a manifold. We also can make the substitutions $$\eta\rightarrow g$$ and $$\partial\rightarrow\nabla$$, which yields the correct (there are exceptions) GR equations.

2. This is required for the geodesic equation to be obtainable from a variational principle because it implies the Christoffel symbols are symmetric. This condition is relaxed in certain theories such as Einstein-Cartan theory or string theory.

3. Simply put, we need this equation to fix the constants in Einstein's equation.

All treatments of GR use the Principle of Equivalence. Weinberg's treatment especially so. The reason for this has to do with his background as a physicist. Weinberg was one of our greatest particle physicists. His dream was to write a coherent quantum field theory for gravity. In his mind, $$g_{\mu\nu}$$ being called the metric tensor is an "antiquated" term left over from when Einstein learned differential geometry from his friend Grossmann and Riemann & co.'s old papers.$$^1$$ In Weinberg's mind, $$g_{\mu\nu}$$ is just the graviton field, and any connection to geometry is purely formal.$$^2$$ In texts such as Carroll, Straumann or Wald, they use the EP to make the connection $$\tag{1}\text{Equivalence Principle}\implies\text{Spacetime is a manifold}$$ From that point on, spacetime being a manifold is assumed. Weinberg, however, was of the opinion that gravity had nothing to do with geometry and manifolds and this mathematical description was a pure formality. He has to stress the EP because philosophically he didn't accept (1).

$$^1$$ See the first paragraph of section 6.9.

$$^2$$ See, for instance, page 77, where he calls the geodesic equation a mere formal analogy to geometry.

• Don't you need also that the connection is compatible with the metric?
– MBN
Commented Feb 11, 2015 at 20:23
• @MBN: Once we have symmetric Christoffel symbols, we can construct a metric connection. The more general statement would be to say that the manifold is pseudo-Riemannian. On a pseudo-Riemannian manifold, the Levi-Civita connection is unique. Commented Feb 11, 2015 at 21:01
• A question regarding Weinberg's philosophical point of view; does his take on relativity yield any different results, or change anything? Commented Feb 11, 2015 at 22:36
• @PhysicsLlama: Not that I can think of. The consequences are mostly academic in nature. Despite G&C being a graduate text, the math is really quite simple because he views the more advanced differential geometric notions of charts, homeomorphisms, etc. as formal nonsense. (Weinberg was a shut up and calculate purist.) There is one thing that strikes out though: despite G&C being written in 1972, black holes are nowhere to be found in the book. A glance at the table of contents reveals a one-page section on the Schwarzschild singularity. (Cont. in next comment.) Commented Feb 11, 2015 at 22:48
• @PhysicsLlama: In this section, he acknowledges that Hawking and Penrose had shown, using topological methods, that trapped surfaces were feasible. Weinberg, however, expresses doubt. For one, keep in mind that in 1972, there was no experimental evidence for black holes. I have a personal theory on the matter. Since he didn't think spacetime was a manifold, he certainly wouldn't think topology could be used to justify anything in GR. Also, Weinberg probably didn't even know any topology! I don't think topology was introduced in QFT until the mid-70s. Commented Feb 11, 2015 at 22:57

I think one can enter a dispute regarding the notion of "accepted" but the idea is that General Relativity is successfully described by a Pseudo-Riemannian Manifold, subject to Einstein Equations, with free-falling objects following geodesics. Now you look for a set of axioms that give you this structure. One such set, although not entirely rigorous, is found in a paper by Ehlers, Pirani and Schild called the "The geometry of free fall and light propagation". I'll give you a brief discussion of the contents.

Start with two principles, (1) the Einstein Equivalence Principle and (2) the Finiteness of the velocity of light. The first one says that objects in free-fall in the gravitational field are in inertial movement and the second one says that not only light traves at a finite velocity but nothing goes faster than it.

Putting in another words principle (1) dictates that the trajectory of free-fall is as "straight" a line, from the point of view of an observer, as a constant velocity trajectory is in Newtonian physics. Principle (2) establishes that since everything travels at finite velocity there is a causal relation between events, namely if two events are have a spatial distance bigger than the velocity of light times the time separation they cannot have a cause-effect relationship, which of course implies in the relativity of simultaneity and other stuff from special relativity.

In more specific sense, principle (1) gives you a set of "straight lines", i.e. a set of geodesics, while principle (2) gives you a set of causal relations between events. In the paper by Ehlers, Pirani and Schild they call this two structures (1) a Projective Structure and (2) a Conformal Structure. Then they show that this two, with the assumptions that they are compatible and clocks behave in a reasonable way. imply the existence of a unique Lorentzian metric and Riemann tensor, together with the interpretation of geodesics and light-cones. All that's left if to require that the geodesics deviation be compatible with the one from Newtonian theory to obtain the Einstein Equations.

They do provide a set of axioms that address every part of the assumptions, but it all can be traced back to these two principles, Einstein Equivalence and finite velocity of the propagation of light.

As a remark, you may note that this idea is very different from what Weinberg exposes, namely that geometry is not fundamental in this description, but, as he himself says in page 147, this is an heterodox point of view not subscribed to by general relativists. On the other hand it is, as far as I'm aware, the mainstream point of view in String theory.

• How do those axioms give rise to the notion of a pseudo-Riemannian manifold? (Not challenging you, I just don't have access to the paper.) Commented Feb 11, 2015 at 22:21
• No problem. As I said it is not entirely rigorous, but the idea is this: the conformal structure dictates which events are space-like, time-like or null. In others words it defines a conformal metric, i.e. a set of metrics together with a equivalence relation $g_{\mu\nu} \equiv h_{\mu\nu}$ if $g_{\mu\nu}=\Omega^2h_{\mu\nu}$. In other words the conformal structure picks a metric only up to a positive factor. Different functions $\Omega$ give different geodesics. By choosing the projective structure you fix the scale factor. Then you get the whole lorentzian metric. Commented Feb 12, 2015 at 13:15
• Forgot to mention, there are some extra assumptions to guarantee uniqueness. The conformal plus projective structures must be compatible (the null geodesics must contains points with null separation), an even then you need to further assume that a vector when parallel transported to the same point along different curves althought may change direction it does not change norm. Commented Feb 12, 2015 at 13:21

In Einstein's decisive article on the "Foundations of General Relativity" (1916) we can find a statement which at least has the appearance and may be understood as if it were an axiom, namely:

"All our space-time verifications invariably amount to a determination of space-time coincidences {... such as ...} meetings of two or more {...} material points."

Arguably, this entails that

• space-time coincidences are thought as identifiable and distinguishable by identifying who had taken part, case by case; and therefore

• participants (such as "material points") being thought as individually identifiable and recognizable, and

• each participant being thought capable, at least in principle, to distinguish which of his perceptions he collected in coincidence (i.e. together), and which instead successively.

The notion of "space-time coincidence" is in more current terminology of course also known as "spacetime event"; at least as far as referring to individuated events.

If the quoted statement were taken rigorously as an axiom, all remaining "space-time terminology" of general relativity should therefore be definable, and indeed explicitly defined, in terms of (relations between) space-time coincidences, or therefore in terms of identifiable participants.

(To give some examples of which notions then ought to be defined accordingly:

• of any three distinct events in which at least one particular participant had been participating: how to measure which one event had been "chronologically between" the two others, cmp. "causal relations";

• of any participant, between any two events in which he took part: how to measure whether or not he had been "free" during this particular stretch of his existance, cmp. the prerequisits of "Ehlers-Pirani-Schild";

• of any two "free" participants: how to measure whether or not they had been "sitting still" wrt. each other; cmp. W. Rindler's characterization of an "inertial frame as a set of (identifiable) point particles sitting still wrt. each other", in turn as a prerequisit for conceiving of "the equivalence principle" even in its basic geometric version. )

However, apparently Einstein did not strictly persue this particular axiomatic approach; and, to my knowledge, it is still largely left to be persued.